41. Recommended Cryptography Books: Prerequisites For Chinese Remainder Theorem chinese remainder theorem Applications in Computing, Coding, Cryptography Ding,C. / Pei, D. / Salomaa, A.. 1996. 213 pages. Categories Mathematics. http://www.youdzone.com/cryptobooks_9810228279_prereqs.html

Prerequisites for Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography Ding, C. / Pei, D. / Salomaa, A.. 1996. 213 pages. Categories: Mathematics The book covers some fun stuff, like the Chinese using the CRT to compute the numbers of different size blocks required for building the Great Wall of China, and of Chinese generals that had their soldiers line up in group of which the general merely counted the remainder to keep their true numbers secret. But on the whole, this book is not for someone with a casual interest in the subject, and definitely requires background in number theory, information theory, and abstract algebra. Not light reading. Recommended prerequisite books: This book: (Read review) Suggested mathematical background in: - Linear Algebra Suggested computer language experience: N/A

42. Chinese Remainder Theorem Divide your number by 3 and enter the remainder in the first box below on the left. Divideyour number by 5 and enter the remainder in the box in the center. http://www.doylestown.com/chinese.html

Choose a number between 1 and 104, maybe 32, maybe 71, or whatever. This will be called 'your number'. Divide your number by 3 and enter the remainder in the first box below on the left. For example, if your number is 17, divide 17 by 3 and get 5 with a remainder of 2. Enter 2 in the box. Divide your number by 5 and enter the remainder in the box in the center. Divide your number by 7 and enter the remainder in the box on the right. After all three remainders have been entered, push the OK button on the right. Page last updated January 16, 2002. - - Doylestown Home Page Email Charlie or call 330.658.6076

43. Chinese Remainder Theorem Can Be Seen See here for the ps.gz version. Andrzej Solecki's cecinestpasunepeep Show. proudlypresents 5 as m, 4 as n and x as x in the great. chinese remainder theorem. ! http://www.ilhadamagia.com.br/~andsol/english/mat/china.html

See here for the ps.gz version Andrzej Solecki's cecinestpasunepeep Show proudly presents as m as n and x as x in the great Chinese Remainder Theorem index support file

44. Solving Congruences: The Chinese Remainder Theorem Solving Congruences The chinese remainder theorem. This is done by the Chinese RemainderTheorem, socalled because it appeared in ancient Chinese manuscripts. http://www.math.okstate.edu/~wrightd/crypt/lecnotes/node21.html

Next: Challenges! Up: Cryptology Class Notes Previous: Square roots Solving Congruences: The Chinese Remainder Theorem In considering the problem of finding modular square roots, we found that the problem for a general modulus m could be reduced to that for a prime power modulus. The next problem would be how to piece the solutions for prime powers together to solve the original congruence. This is done by the Chinese Remainder Theorem, so-called because it appeared in ancient Chinese manuscripts. A typical problem is to find integers x that simultaneously solve It's important in our applications that the two moduli be relatively prime; otherwise, we would have to check that the two congruences are consistent. The Chinese Remainder Theorem has a very simple answer: Chinese Remainder Theorem: For relatively prime moduli m and n , the congruences have a unique solution x modulo mn Our example problem would have a unique solution modulo It's better than this; there is a relatively simple algorithm to find the solution. Since all number theory algorithms ultimately come down to Euclid's algorithm, you can be sure it happens here as well. First let's consider an even simpler example. Suppose we want all numbers

45. Numlib::ichrem -- Chinese Remainder Theorem For Integers numlibichrem chinese remainder theorem for integers. Introduction.numlibichrem (a,m) returns the least nonnegative integer http://www.mupad.com/doc/de/numlib/ichrem.shtml

numlib::ichrem Chinese remainder theorem for integers Introduction numlib::ichrem (a,m) returns the least nonnegative integer x such that (x-a[i]) mod m[i]=0 for i=1,..,m if such a number exists; otherwise numlib::ichrem (a,m) returns FAIL Call(s) numlib::ichrem(a, m) Parameters a a list of integers m a list of natural numbers of the same length as a Returns either a nonnegative integer or FAIL Related Functions numlib::lincongruence Details The entries in m need not be pairwise coprime. numlib::ichrem (a,m) returns an error if a is not a list of integers or m is not a list of natural numbers or a and m are not lists of the same length. Example 1 Here the moduli are pairwise coprime. In this case, a solution always exists: numlib::ichrem([2,3,2],[3,5,7]) Example 2 Here the moduli are not pairwise coprime, and a solution does not exist: numlib::ichrem([5,6,8],[20,21,22]) FAIL Example 3 Also here the moduli are not pairwise coprime, but a solution nevertheless exists: numlib::ichrem([5,6,7],[20,21,22]) Do you have questions or comments?

46. Chinese Remainder Theorem chinese remainder theorem. Solve the followinglinear congruences x a i (mod m i ) x=. http://people.ucsc.edu/~erowland/crt.html

bg="white"; Chinese Remainder Theorem Solve the following linear congruences x a i (mod m i x

47. Untitled and is the origin of the chinese remainder theorem. The song appeared firstin Sun ZI Suan Jing of the 4th Century chinese remainder theorem. http://www.weizmann.ac.il/~feshtrik/sunzig.html

Is a Chinese folk song hiding the answer to: What is the number that gives when devided by 3 a remainder of 2 when devided by 5 a remainder of 3 when devided by 7 a remainder of 2 and is the origin of the Chinese Remainder Theorem The song appeared first in "Sun ZI Suan Jing" of the 4th Century Chinese Remainder Theorem The Song of Master Sun 3 men walk together for 70 miles 5 plum trees blossom 21 branches 7 persons reunion at June 15 Circulating at a period of 105. Back to Shmuel Shtrikman Home Page

48. Chinese Remainder Theorem AA Jagers. http://wwwhome.math.utwente.nl/~jagersaa/152063/CRS/Main.html

A.A. Jagers A.A. Jagers

49. MathGroup Archive: November 1998 Re: The Chinese Remainder Theorem Re the chinese remainder theorem. Prev by thread Re the Chinese RemainderTheorem; Next by thread Re the chinese remainder theorem. http://forums.wolfram.com/mathgroup/archive/1998/Nov/msg00034.html

January February March April ... Author Index Re: the Chinese Remainder Theorem To mathgroup@smc.vnet.net Subject : [mg14616] Re: the Chinese Remainder Theorem From hay@haystack.demon.co.uk Date : Wed, 4 Nov 1998 13:46:47 -0500 References Sender owner-wri-mathgroup@wolfram.com Prev by Date: Simple cure for Mac crash/password problems Next by Date: Re: Options using Print Prev by thread: Re: the Chinese Remainder Theorem Next by thread: Re: the Chinese Remainder Theorem

50. MathGroup Archive: November 1998 Re: The Chinese Remainder Theorem Re the chinese remainder theorem. To mathgroup@smc.vnet.net; Subjectmg14630 Re mg14597 the chinese remainder theorem; From http://forums.wolfram.com/mathgroup/archive/1998/Nov/msg00055.html

January February March April ... Author Index Re: the Chinese Remainder Theorem To mathgroup@smc.vnet.net Subject : [mg14630] Re: [mg14597] the Chinese Remainder Theorem From mwmak@ix.netcom.com Date : Wed, 4 Nov 1998 13:47:00 -0500 Organization : Harmonic Systems References 199811020651.BAA08838@smc.vnet.net. Sender owner-wri-mathgroup@wolfram.com References the Chinese Remainder Theorem From: Prev by Date: Re: Mathematica 3.0 and QuickTime Next by Date: Re: Too Tiny Fonts! in fractions, summations, etc. Prev by thread: the Chinese Remainder Theorem Next by thread: Re: the Chinese Remainder Theorem

51. Bookmarks2 For Patrick Reany generalizations Rings. chinese remainder theorem. Reany's Heuristicsof chinese remainder theorem Math 5410 chinese remainder theorem http://www.ajnpx.com/html/Bookmarks2.html

Other Links Mathematics Natural Philosophy Patrick Reany's Home Page Bookmarks2 for Patrick Reany Computer Utilities Sysinternals Freeware Computer security CERT Coordination Center L0pht Heavy Industries 403-Security.org HACKER WHACKER Remote Computer Network Security Scan ... NT Security - Frequently Asked Questions Computer Networking IP address index - network numbers, network names and identities Windows Win ME About Windows 2000 - NT WinNT About Windows 2000 - NT FixWindows.com - The Windows Troubleshooting Site General Troubleshooting Strategy Newport Beach, California My Newport Beach Page The Newport Beach Conference And Visitors Bureau Newport Hotels : Hyatt Newporter : Newport, California California Vacation Rentals ... TravelHero.com - BEST WESTERN BAY SHORES INN - Reservations Philosophy Natural Philosophy Positivism histphsc Philosophy since the Enlightenment, by Roger Jones: Foucault, Hegel, Marx, Nietzsche, Wittgenstein Post-Positivism Elucidated Skeptic's Dictionary: logical positivism ... Tel-Aviv University Philosophy Department: Philosophy Sites on the Internet Networking High Speed Connections cabling 3M Volition Network Cabling Solutions: the leader in fiber optic cable networking Email servers Mail Servers - ServerWatch Mail Servers Tutorials computer related Namezero SQL Namezero Programming HTML Java Namezero C++ Visual Basic Scripts Mathematics Flow Charts Math Flow Charts Recursive Functions, Flow Charts, And Algolic Programs

52. An English-Persian Dictionary Of Algorithms And Data Structures chinese remainder theorem, ÞÖíåí ÈÇÞíãÇäÏåí íäí.algorithm. íԝäåÇÏ ÊÑÌãåí ÌÏíÏ. Definition http://ce.sharif.edu/~dic-ads/d.php?r=Chinese remainder theorem.1.8

53. Chinese Remainder Theorem next up previous Next Up Previous chinese remainder theorem. Find integersx that leave remainders 2, 3, 2 when divided by 3, 5, 7, respectively. http://ranger.uta.edu/~cook/aa/lectures/l23/node22.html

Next: Up: Previous: Chinese Remainder Theorem Find integers x that leave remainders 2, 3, 2 when divided by 3, 5, 7, respectively. [Sun-Tsu, 100 A.D.] Theorem 33.27 Let , where n i are pairwise relatively prime and consider the correspondence where , and a i = a mod n i for i = 1, ,k. Next: Up: Previous:

54. Chinese Remainder Theorem (33.27 Cont.) next up previous Next Up Previous chinese remainder theorem (33.27cont.). If and. Then (a+b) mod n ((a 1 + b 1 ) mod n 1 , , (a http://ranger.uta.edu/~cook/aa/lectures/l23/node23.html

Next: Up: Previous: Chinese Remainder Theorem (33.27 cont.) If and Then (a+b) mod n a b ) mod n a k b k ) mod n k (a-b) mod n a b ) mod n a k b k ) mod n k ab mod n a b mod n a k b k mod n k From (a mod n i , a mod n k From for i = 1, , k (mod n) Next: Up: Previous:

55. Chinese Mathematics : Rebecca And Tommy The chinese remainder theorem (TaYen). It was not until 1247 thatQin Jiushao (c 1202-1261) published a general method for solving http://www.roma.unisa.edu.au/07305/remain.htm

The Chinese Remainder Theorem (Ta-Yen) It was not until 1247 that Qin Jiushao (c 1202-1261) published a general method for solving systems of linear congruence's in his book called ' Shushu jiuzhang (Mathematical Treatise in Nine Sections)' (Katz, 1992, p188). A book clearly influenced by the old chiu chang suan shu , as were a majority of Chinese mathematical works. Before this time only specific problems had been solved, by people such as Shu Zi (late third century). This method became known as the Ta-Yen. The basic format of problems it was to solve was ; N = a(mod b) = c(mod d) = ... This meant find N such that when divided by b gives a remainder of a and when divided by d gives a remainder of c. Throughout this page we will use the example N = 10(mod 12) = 0(mod 11) = 0(mod 10) = 4(mod 9) = 6(mod 8) = 0(mod 7) = 4(mod 6) In simple terms the method goes like this : 1. Find the least common multiple of the moduli. In our example the moduli are 12,11,10,9,8,7 and 6. How was this done? Reduce all moduli to a multiplication of prime numbers or their powers, unless they are already prime or a power of a prime alone. x x 3, 11 = 11 (already a prime), 10 = 2

56. Chinese Remainder Problem Unfortunately, Problem 26 is the only problem that illustrates thechinese remainder theorem in the Sun Tzu Suan Ching. As such, we http://www.math.sfu.ca/histmath/China/3rdCenturyBC/CRP1.html

Chinese Remainder Problem - The Beginning. What is it? These particular kinds of mathematical problem falls in the category of indeterminate analysis. Usually, it appears in the form as such (in modern notation): N = m1x + r1 N = m2y + r2 N = m3z + r3 Or in modern number theory notation: N r1 (mod m1) N r2 (mod m2) N r3 (mod m3) Aside: Writing N r1 (mod m1) [this means N is congruent to r1 modulo m1] means that N divided by m1 leaves r1 as the remainder. The goal here is to find the smallest positive integer satisfying the congruences states above. Origins. Now that you know what a Chinese Remainder Problem is, you must be wondering why or what has this particular kind of problem to do with Chinese Mathematical History. The reason why it is called the Chinese Remainder Problem is because the earliest versions of these congruence problems occured in early Chinese mathematical works. The earliest of such works that contains the Chinese Remainder Problem is the Sun Tzu Suan Ching (also known as Sunzi suanjing) written in approximately late third century by Sun Zi . Problem 26 (also known as the problem of Master Sun) in the third volume of the Sun Tzu Suan Ching offers the earliest recorded Chinese Remainder Problem. Problem 26 is as stated below: "We have a number of things, but we do not know exactly how many. If we count them by threes we have two left over. If we count them by fives we have three left over. If we count them by sevens we have two left over. How many things are there?" (Quoted from Sun Tze Suan Ching).

57. Www.cs.nott.ac.uk/~lad/challenges/IWC008.txt chinese remainder theorem Summary Needs many lemmas(some of whose proofs are also challenging) An Existential Witness has to http://www.cs.nott.ac.uk/~lad/challenges/IWC008.txt

Springer-Verlag.Chinese Remainder Theorem - Summary: Needs many lemmas (some of whose proofs are also challenging) An Existential Witness has to be Provided Definitions: allcongruent (num, list) -> bool allcongruent(x, nil) = true allcongruent(x, y::z) = allcongruent(x, z) and (rem(x, first(y))=rem(second(y), first(y))) allpositive - list -> bool true if all members of the list are greater than 0. allprime2 - list -> bool allprime2(nil) = true allprime2(y::z) = prime2list(x, first(y)) and allprime2(x, z) prime2 is true iff its argumens are relatively prime prime2list (num, list) -> bool prime2list(x, nil) = true prime2list(x, y::z) = prime2(x, first(y)) and prime2list(x, z) products1 - product of list. rem - returns the remainder of x by y Theorem: forall l:(list (pair nat)). exists x:nat. (allpositive(l) and allprime2(l)) -> allcongruent(x,l) forall l:(list (pair nat)). forall x,y:nat. ((allpositive(l) and allprime2(l)) and allcongruent(x,l) and allcongruent(y,l)) -> (mod (x - y),products(l)) Comments Actually two proof, existance and uniqueness. Proved in RRL by Zhang and Hua and there is a good deal of comment of the proof in their CADE-11 paper on the subject. They provide by hand the existential witness needed for the existance part of the proof. Source Proving the Chinese Remainder Theorem by the Cover Set Induction, H. Zhang and X. Hua, CADE-11, D. Kapur (ed), 1992.

58. 0.5.14 Chinese Remainder Theorem Prime Number Up 0.5 Miscellaneous Algorithms Previous 0.5.13 Horner'sRule 0.5.14 chinese remainder theorem. Scott Gasch 199907-09 http://www.fearme.com/misc/alg/node139.html

Search Next: 0.5.15 Large Prime Number Up: 0.5 Miscellaneous Algorithms Previous: 0.5.13 Horner's Rule 0.5.14 Chinese Remainder Theorem Scott Gasch

59. [quant-ph/9911051] Large Numbers, The Chinese Remainder Theorem, And The Circle 201250 GMT (12kb) Large Numbers, the chinese remainder theorem,and the Circle of Fifths. Authors SA Fulling Comments 9 pages http://arxiv.org/abs/quant-ph/9911051

Quantum Physics, abstract quant-ph/9911051 Large Numbers, the Chinese Remainder Theorem, and the Circle of Fifths Authors: S. A. Fulling Comments: 9 pages; Plain TeX with vanilla.sty and pictex.tex macros This is a pedagogical article cited in the foregoing research note, quant-ph/9911050 Full-text: PostScript PDF , or Other formats References and citations for this submission: SLAC-SPIRES HEP (refers to , cited by , arXiv reformatted); CiteBase (autonomous citation navigation and analysis) Links to: arXiv quant-ph find abs

60. Chinese Remainders The chinese remainder theorem is not particularly easy to understand it justsays that under certain conditions (ie the divisors have no common factors http://www.delphiforfun.org/Programs/chinese_remainders.htm

Home Introduction About Delphi Feedback ... Site Search (bottom of page) Available Now (Click a level below to expand program list if it is collapsed - MS Internet Explorer only) Beginners 10 Easy Pieces Bouncing Ball A Card Trick? Cards #1 ... Multiple of 12? Intermediate 15 Puzzle #1 The 9321 Problem Aliquot Sums (Friendly If not Perfect) Alphametics ... Word Ladder Advanced 15 Puzzle #2 Akerue Brute Force Countdown Plus! ... WordStuff 2 Contact Feedback Send an e-mail with your comments about this program (or anything else). Search WWW Search delphiforfun.org Problem Description What is the smallest number that can be divided by 6, leaving a remainder of 5; by 5 leaving 4; and by 4 leaving 3? Source: The Mensa (c) Puzzle Calendar for Oct 18. 2001; Here's an introduction to Chinese Remainder problems. They're called "Chinese Remainder" because the problem and the theorem which defines when they can be solved were both known to the early Chinese scholars. The earliest known example of this type of problem was published in the 3rd, 4th or 5th century AD by Chinese scholar, Sun Zi, in a book titled "

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